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One of my players is playing an Artificer, and has just reached level 2. Reading the infusion rules, we are both a little confused as they seem to contradict.

The description of the Infuse Item feature seems to say that Infusions last indefinitely until either the Artificer dies, places the same infusion on a different item, or loses the knowledge of that infusion. I take to mean that over time an artificer can have every infusion running on multiple items.

For instance, the artificer knows 6 infusions. Day 1, he casts 1 and 2 on his armour and a weapon; day 2, he casts 3 and 4 on a different party member's equipment; and day 3, he casts the final 2 on a different party member.
Is this the right application of the rules? To me, it makes sense. Otherwise, what is the point of having a class that makes magic items?

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The Artificer class table lists the limit on the number of infused items you can have at any given time.

The rules for infusions in the description of the artificer's Infuse Item feature state (TCoE, p. 12-13; ERftLW, p. 57; WGtE, p. 79):

You can infuse more than one nonmagical object at the end of a long rest; the maximum number of objects appears in the Infused Items column of the Artificer table. [...] If you try to exceed your maximum number of infusions, the oldest infusion immediately ends, and then the new infusion applies.

Consult the Artificer table to see how many items you can have infused at once.

For example, at 5th level, the table says you can infuse up to two items. So you can infuse two items and no more than two items. If you try to infuse a third item, the infusion on the first item ends.

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Your example has some misconceptions

The example you provided have some mistakes on it. In that example the artificer have 6 infusions known, which would mean he would be around level 6~9, according to the Artificier Table (pg. 10 of TCoE).

Although he knows 6 infusions, the artificer in that range of level can have only 3 infused items at a time, according the same table. That means you could change between those 6 infusions, but at that range of level you could not have the 6 infusions at the same time.

Also, you do not "cast" the infusion and infusing an item does not consume any spells slots. You also does not need to "maintain" an infusions, like a concentration or by repeatedly infusing the same item. The only resource you spend in infusing an item would be the amount of items you can still infuse.

Other thing worth to be pointed out is that you can infuse more than one object during a long rest, and the maximum it's the maximum infused items you can have. That means that under your example, where you know 6 infusions and can have 3 infused items, you could infuse 3 items during a long rest, choosing among the 6 infusions you know.

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You know a certain number of infusions, and can have half that many running at any given time.

The idea here is that you know the formulas to a number of quasi-permanent magical objects, but you can only have half as many running at once, and you can only have one instance of each formula running at a time. Each morning you can swap out which infusions you're using, but you always have to respect the maximum you can have at once.

As an example, at 2nd level, your artificer knows 4 infusion formulas, but can only have 2 running concurrently. Suppose he's chosen Enhanced Weapon, Enhanced Defense, Replicate Item (Wand of Secrets), and Replicate Item (Bag of Holding).

On day 1, they're setting out on a new adventure, and don't really know what's going to happen, so he uses Enhanced Weapon and Enhanced Defense on the fighter's gear. On day 2, they've found the entrance to an underground complex, so he decides to leave the armor enhancement alone, but turns off the weapon enhancement and instead creates a Wand of Secrets, to better search for hidden doors and traps. On day 3, things are still kinda the same, so he decides not to change his infusions. On day 4, they've cleared the dungeon and have a lot of loot to take home, so he allows the Enhanced Defense infusion to lapse and instead makes a Bag of Holding to help more easily carry their stuff.

You only get two items at a time at 2nd level, but you can pick which ones are more useful to you for any given day.

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